THEOREM 3 
(compare with [1]}, Theorem 1)
Let \( F,C,X\)  and \( N(F)\)  be as in Lemma REF . Assume that for any \(x \in C\)  there exists \( h \in N(F)\)  such that \( h(x) \in F.\)  Then there exists \( g \in N(F)\)  such that for any \(x \in C\)  \(g(x) \in F.\)  In particular, \(r(x) \in R_F(x)\)  for any \( x \in C,\)  where \(r\)  is a maximal element from Lemma REF . If \(C =X, \)  then \(F\)  is an existence set in \(X.\)

